Function Expression :
\[f(x)=\frac{x^3+6x^2+2}{2(x+2 )^3} \]Domain
\[\left]-\infty, -2\right[ \cup \left]-2, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = \frac{1}{2} \]\[\lim_{x \overset{<}{\rightarrow-2} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow-2} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = \frac{1}{2} \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 \left(x + 2\right)^{3}} \cdot \left(3 x^{2} + 12 x\right) - \frac{3 \left(x^{3} + 6 x^{2} + 2\right)}{2 \left(x + 2\right)^{4}} \]\[f^{\,\prime}(x)=\frac{3 \cdot \left(4 x - 1\right)}{x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16} \]
\[ \]
Integral
\[F(x) = \frac{x}{2} + \frac{12 x + 15}{2 x^{2} + 8 x + 8} \]Sign Table
Variation Table
Plot
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